Diagnostic apparatus

ABSTRACT

Apparatus and method for determining a likely cause or the likelihood of the occurrence of a cause of one or more effects, in which training data relating to previously identified relationships between one or more causes and one or more effects is used to learn the cause and effect relationship. A number of primary and secondary reference points are chosen in the input space created by belief values representing the strength of effect. A Lagrange Interpolation polynomial (or other function representing the cause and effect relationship) and a weight value is associated with each of the said reference point. Weight values associated with primary reference points are considered as independent variables (primary weight values) and other weight values, which are associated with secondary reference points (secondary weight values), depend (preferably, but not necessarily, linearly) on one or more primary weight values. Belief value in the occurrence of likely causes of one or more given effects can be determined using this method or apparatus.

[0001] This invention relates to an apparatus and method for assistingin diagnosing cause and effect in many different circumstancesthroughout various different disciplines.

[0002] In recent times, various disciplines, including the manufacturingindustry and at least some branches of the medical field, have comeunder increasing pressure to increase yield, productivity and profits(or reduce costs or overheads). As such, there is an increasing need toreduce costs and reach the required result as quickly as possible, forexample, in the manufacturing industry, the required result may be themanufacture of a batch of products which are all of optimum quality, andin a branch of the medical field, the required result might be thecorrect diagnosis of a complaint or disease in a patient (i.e. in bothcases, to get it “right first time”).

[0003] In the case of the manufacturing industry, manufactured productsare usually tested for quality, and sub-standard units are rejected.When a unit or set of units is rejected, the fault or faults aregenerally noted and it is highly desirable to establish the reason forthe occurrence of the faults so that any problems can be rectified andthe chances of manufacturing sub-standard products thereafter can beminimised. Such diagnosis is usually performed by experts in the field,who have acquired a fundamental understanding over the years of causeand effect relationships, which often influences the occurrence ofrejected product units. Similarly, a patient will usually visit ageneral practitioner with one or more symptoms, from a description ofwhich the doctor must try and diagnose the problem, based on currentmedical knowledge and his experience in the field. Thus, in general,cause and effect relationships are learned by experts in a particularfield and applied in the diagnosis of problems when they occur in thatfield.

[0004] However, this manual diagnosis procedure is time consuming and,therefore, increases costs and reduces productivity and yield. Further,when an expert leaves a particular place of employment, his expertise isalso lost to the employer.

[0005] Attempts have been made in the past to relate belief values,representing the strength of an effect, to a belief value whichquantifies the extent of occurrence of an associated cause, using eithermultivariate regression analysis methods or neural network relatedmethods. If the belief values in ‘p’ manifestations associated with acause ‘c’ are represented by ‘p’ variables ξ¹, ξ², ξ³, . . . ξ^(j), . .. ξ^(p) (as shown in FIG. 1), then using the multivariate linearregression analysis method [1] or single layered feed forward neuralnetwork method [2], the belief value, which quantifies the extent ofoccurrence of cause ‘c’, is given by the following equation:

Belief in cause=w ₀ +w ₁ξ¹ +w ₂ξ² +w ₃ξ³ + . . . +w _(j)ξ^(j) + . . . +w_(p)ξ^(p)  (1)

[0006] where w_(j), j=0 to p are referred to as either regressioncoefficients (in a regression analysis context) or weights (in a neuralnetwork context). These coefficients, or weights, generally beingconsidered as independent variables, are mostly determined using leastsquare minimisation techniques. This is achieved by comparing the beliefvalue in the cause calculated by equation (1) with a previously knownvalue for the same inputs.

[0007] Multi-layered feed forward neural network techniques, includingradial basis functions [2], and also a range of methods proposed in thefamily of intrinsically linear, multivariate regression analysis [1],generalise equation (1) in the following way:

Belief in cause=w ₀ +w ₁ z ₁ +w ₂ z ₂ +w ₃ z ₃ + . . . w _(i) z _(i) + .. . +w _(m) z _(m)  (2)

[0008] where z_(i) (i=1 to m) represents a function of ξ_(j), (j=1 to p)

[0009] Different methods and apparatuses may use different functions(z_(i)) ranging from simple linear polynomials to higher order,non-linear polynomials, logarithmic or exponential functions.

[0010] Although, these methods have been employed to associate beliefvalues in causes to belief values in effects, the methods have two majorlimitations:

[0011] 1. Physical interpretation can not be assigned to either weightsw_(i) or functions z_(i), as given in equation (2), which makes it verydifficult to gain any physical insight into the cause and effectrelationship.

[0012] 2. The number and the type of function used to define z_(i) arenot unique and are determined by a trial and error method.

[0013] These limitations constrain the applicability of existing methodsto relate belief values, which quantify the occurrence of a cause, withbelief values representing strength of effects.

[0014] The inventors have now devised an arrangement that overcomes thelimitations/problems outlined above and seeks to provide a generic toolto relate belief values in causes and effects for use in many differentfields, and industries, for diagnosing problems occurring therein.

[0015] Thus, in accordance with the first aspect of the presentinvention, there is provided apparatus for determining a likelihood ofoccurrence of a cause of one or more effects, the apparatus comprisingmeans for receiving and/or accessing training data relating topreviously-identified relationships between one or more causes and oneor more effect means, for defining one or more functions representativeof said relationships, said function(s) being in the form of polynomialswhich define quantified mappings between said one or more causes and oneor more effects, each of said polynomials being associated with each ofa plurality of respective reference points, at least some of which havea weight value assigned thereto, at least one of said weight valuesbeing an independent variable (“primary weight”) and at least one ofsaid weight values being dependent on one or more of said primaryweights “(secondary weight”), and means for determining a likelihood ofoccurrence of one or more causes of one or more given effects using saidmappings.

[0016] Also in accordance with the first aspect of the presentinvention, there is provided a method of determining a likelihood ofoccurrence of a cause of one or more effects, the method comprising thesteps of receiving and/or accessing quantifying data relating topreviously-identified relationships between one or more causes and oneor more effects, defining one or more functions representative of saidrelationships, said function(s) being in the form of polynomials whichdefine quantified mappings between said one or more causes and one ormore effects, each said polynomials being associated with each of aplurality of respective reference points, at least some of which have aweight value assigned thereto, at least one of said weight values beingan independent variable (“primary weight”) and at least one of saidweight values being dependent on at least one of said primary weights(“secondary weight”), and determining a likelihood of occurrence of oneor more causes of one or more given effects using said mappings.

[0017] Thus, the present invention not only provides a method andapparatus for automatically providing diagnostic information relating toone or more likely causes of one or more given manifestations oreffects, thereby reducing the input required by expert personnel oncesufficient training data has been entered, where human input canotherwise be time consuming and is, of course, prone to error as it isgenerally based only on the opinion of very few experienced personnel;but also significantly reduces the number of independent weight valuesrequired to facilitate the diagnostic process, thereby reducing thequantity of training data required to a practical level when a largenumber of effects or manifestations of a cause are involved.

[0018] The present invention provides a method and apparatus forcalculating a belief value which quantifies the extent of occurrence ofa cause, given belief values which quantify the occurrence, ornon-occurrence, of associated affects of the cause. Examples of effectsof a cause, which are indicative but certainly not exhaustive, are“symptoms” shown by patients in the medical domain, “defects” occurringin components in manufacturing industry or “effects” as generally meantin any “cause and effect” diagram.

[0019] The belief value quantifying the occurrence, or non-occurrence,of an effect associated with a particular cause is preferably normalisedbetween +1 to −1 or 1 to 0 respectively. This belief value may also beinterpreted as being the strength of the effect. The belief value, whichquantifies the extent of occurrence of the cause under consideration, ispreferably also normalised from zero to unity or −1 to +1 to representnon-occurrence and occurrence, respectively.

[0020] The present invention also allows a meaningful physicalinterpretation to be assigned to each and every said weight value(primary and secondary weights) in the sense that a weight value at aposition described by belief values, which represent the strength ofassociated manifestation, is nothing but the output value i.e. a beliefvalue representing the extent of occurrence of the cause given thestrengths of the associated effects.

[0021] In a preferred embodiment of the present invention, the apparatuscomprises a means of producing a multi-dimensional hyper-surfacerepresenting belief values in the occurrence/non-occurrence of a cause.The number of dimensions of the hyper-surface is equal to the number ofinput nodes representing the effects of a cause. The order of thehyper-surface along each dimension is determined by the order of thepolynomial, preferably a Lagrange Interpolation Polynomial, used alongthe said dimension. The first order (or linear) Lagrange InterpolationPolynomial is defined by two reference points. Second order or quadraticLagrange Interpolation Polynomials require three reference points.Similarly, an n^(th) ordered Lagrange Interpolation Polynomial willrequire (i+1) reference points along the given dimension. The apparatusor method of calculating reference points including primary referencepoints and then associating a weight value along with the LagrangeInterpolation Polynomial at each reference point is described in moredetail below.

[0022] The training data is preferably made up of one or more trainingfiles, the or each file comprising an input vector, storing beliefvalues representing strengths of all associated effects, and itscorresponding desired output vector, storing belief values, whichquantify the extent of occurrence of the corresponding cause.

[0023] An embodiment of the present invention will now be described byway of example only, followed by sample numerical calculation and withreference to the accompanying drawings, in which:

[0024]FIGS. 1A and 1B is a schematic illustration of a manifestationcause relationship;

[0025]FIG. 2A to 2E are graphical representations of some generalone-dimensional cause and effect relationships;

[0026]FIG. 3 is a schematic diagram illustrating a two effect—one causerelationship; and

[0027]FIG. 4 is a schematic diagram illustrating a two dimensional spacedescribing ξ¹, and ξ² axes.

[0028] In order to facilitate the description of an exemplary embodimentof the present invention, a number of general one-dimensional cause andeffect relationships associating belief values (based on the networkillustrated in FIG. 1B) will first be illustrated. FIGS. 2A-2E, showpossible examples of the variation in belief values, representing theextent of occurrence of cause (output value), with respect to the beliefvalues, representing the strength of one of the associated effect.

[0029] Referring to FIG. 2A of the drawings, a linear variation inbelief values is shown, in which when the belief value representing thestrength of effect is at its minimum, the belief value representing theextent of occurrence of the related cause is also at its minimum. As thestrength of effect increases, the belief value in the occurrence ofcause also linearly increases.

[0030] Referring to FIG. 2B of the drawings, a quadratic variation ofcause and effect is shown in which when the belief value representingthe strength of the effect is at its minimum, then the belief value inthe occurrence of the related cause is also at its minimum. As thestrength of the effect starts to increase, the belief value in theoccurrence of the corresponding cause also starts to slowly increase. Asthe strength of the effect increases to about half of its maximum value,so the belief value in the occurrence of the cause suddenly increasesand reaches its maximum value when the strength of the effect reachesits maximum value.

[0031] Referring to FIG. 2C of the drawings, a quadratic variation ofcause and effect is shown which, when the belief value representing thestrength of the effect is at its minimum, then the belief value in theoccurrence of the related cause is also at its minimum. As the strengthof the effect starts to increase, the belief value in the occurrence ofthe cause quickly starts to increase. When the strength of the effect isaround half of its maximum value, the rate of increase in the beliefvalue of the occurrence of the related cause slows down and reaches itsmaximum value when the strength of the effect also reaches its maximumvalue.

[0032] Referring to FIG. 2D of the drawings, a quadratic variation ofcause and effect is shown in which when the belief value representingthe strength of the effect is at its minimum, then the belief value inthe occurrence of the related cause is at its maximum. As the strengthof the effect starts to increase, there is a quick reduction in thebelief value in the occurrence of the cause. As the strength of theeffect increases to about half of its maximum value, the belief value inthe occurrence of the cause slowly decreases and reaches its minimumvalue when the strength of the effect is at its maximum.

[0033] Referring to FIG. 2E of the drawings, a quadratic variation ofcause and effect is shown in which when the belief value representingthe strength of the effect is at its minimum, then the belief value inthe occurrence of the related cause is at its maximum. As the strengthof the effect starts increasing, the belief value in the occurrence ofthe cause slowly starts to decrease. When the strength of the effectreaches around half of its maximum value, the belief value in theoccurrence of the cause starts to decrease quickly and reaches itsminimum when the strength of the efffect is at its maximum.

[0034] It is an object of the present invention to provide a diagnosticarrangement, which not only “learns” from examples, but also quantifiesthe cause and effect relationship, which may be described in thisexemplary embodiment of the invention by a decision hyper-surfaceconstructed by combining Lagrange Interpolation Polynomials. Tworeference points associated with two Lagrange Interpolation Polynomialsand two weight values are required to describe a linear variation in thebelief values as shown in FIG. 2A. Similarly, three reference pointsalong with three Lagrange Interpolation Polynomials and weight valuesare required to describe one dimensional quadratic belief variation asshown in FIGS. 2B-2E. The learning function of the apparatus isequivalent to finding a multi-dimensional hyper-surface, describing thesaid belief variation, which provides a best fit to the training data.

[0035] Thus, an exemplary embodiment of the invention provides a methodfor calculating a belief value, which quantifies the extent ofoccurrence of a cause, given belief values, which quantify theoccurrence or non-occurrence of associated effects of the cause.Examples of effects of a cause are, but not limited to, “symptoms” shownby patients in the medical domain, “defects” occurring in components inmanufacturing industry or “effects” as generally meant in any “cause andeffect” diagram. The belief value ξ, which quantifies the occurrence ornon-occurrence of an effect associated with a particular cause isnormalised between +1 to −1 respectively. This belief value is alsointerpreted to represent the strength of the effect. The belief value,which quantifies the extent of occurrence of the cause underconsideration is also normalised from zero to unity, to representnon-occurrence and occurrence of the said cause respectively.

[0036] The relationship between the belief value, representing thestrength of the effect, and the belief value, representing the extent ofoccurrence of the cause, is assumed to be either linear, quadratic,cubic and so on. The order of the relationship (e.g. one for linear, twofor quadratic, three for cubic etc.) can either be given or calculatediteratively starting from one. To define an n^(th) order relationshipalong one-dimension, (n+1) reference points, equidistant between −1 to+1, are chosen. (If the location of the reference points is notequidistant, these reference points are mapped on to another set ofequidistant reference points.) For each reference point ‘i’, a onedimensional Lagrange Interpolation Polynomial is constructed based onthe following formula: $\begin{matrix}\begin{matrix}{{l_{i}(\xi)} = {l_{k}^{n}(\xi)}} \\{= {\frac{\xi - \xi_{0}}{\xi_{k} - \xi_{0}}*\frac{\xi - \xi_{1}}{\xi_{k} - \xi_{1}}*\frac{\xi - \xi_{2}}{\xi_{k} - \xi_{2}}*\ldots*\frac{\xi - \xi_{k - 1}}{\xi_{k} - \xi_{k - 1}}*}} \\{{\frac{\xi - \xi_{k + 1}}{\xi_{k} - \xi_{k + 1}}*\ldots*\frac{\xi - \xi_{n}}{\xi_{k} - \xi_{n}}}}\end{matrix} & (3)\end{matrix}$

[0037] where,

[0038] n: order of the Lagrange Interpolation Polynomial (one forlinear, two for quadratic, etc.)

[0039] k: A reference point at which the one-dimensional LagrangeInterpolation Polynomial i_(k) ^(n) (ξ) is constructed. k ranges from 0to n.

[0040] i: Ranges from one to total number of reference points i.e.(n+1).

[0041] ξ₀, ξ₁, ξ₂ . . . ξ_(n) are (n+1) equidistant reference pointsfrom ξ₀=−1 to ξ_(n)=+1 with (n+1) corresponding Lagrange InterpolationPolynomials as given by equation (3). The variable ξ, which stores thebelief value representing the strength of the corresponding effect,ranges from −1 to +1. For a “single effect—cause relationship”, theLagrange Interpolation Polynomial is one-dimensional and the referencepoints are drawn along this dimension. If the number of associatedmanifestations for a given cause is ‘p’, the Lagrange InterpolationPolynomial at a reference point ‘i’ will be ‘p’ dimensional and is givenby the following equation:

l _(i)(ξ¹, ξ², ξ³, . . . ξ^(j), . . . ξ^(p))=l _(k) ₁ ^(n) ^(₁) (ξ¹)*l_(k) ₂ ^(n) ^(₂) (ξ²)* . . . *l _(k) _(j) ^(n) ^(_(j)) (ξ^(j))* . . . *l_(k) _(p) ^(n) ^(_(p)) (ξ^(p))  (4)

[0042] where, $\begin{matrix}{{l_{k_{j}}^{n_{j}}\left( \xi^{\quad j} \right)} = {\frac{\xi^{j} - \xi_{0}^{j}}{\xi_{k_{j}}^{\quad j} - \xi_{0}^{\quad j}}*\frac{\xi^{j} - \xi_{1}^{j}}{\xi_{k_{j}}^{j} - \xi_{1}^{\quad j}}*\frac{\xi^{j} - \xi_{2}^{j}}{\xi_{k_{j}}^{j} - \xi_{2}^{j}}*\ldots*\frac{\xi^{j} - \xi_{k_{j} - 1}^{j}}{\xi_{k_{j}}^{j} - \xi_{k_{j} - 1}^{j}}*\frac{\xi^{j} - \xi_{k_{j} + 1}^{j}}{\xi_{k_{j}}^{j} - \xi_{k_{j} + 1}^{j}}*\ldots*\frac{\xi^{j} - \xi_{n_{j}}^{j}}{\xi_{k_{j}}^{j} - \xi_{n_{j}}^{j}}}} & (5)\end{matrix}$

[0043] n_(j): Order of one dimensional Lagrange Interpolation Polynomial(l_(k) _(j) ^(n) ^(_(j)) (ξ^(j))) corresponding to j^(th) dimension thatrepresents the relationship between j^(th) manifestation and the causeunder consideration.

[0044] k_(j): Reference point along j^(th) dimension, at which the onedimensional Lagrange Interpolation Polynomial l_(k) _(j) ^(n) ^(_(j))(ξ^(j)) is evaluated. (k_(j) ranges from 0 to n_(j).)

[0045] ξ₀ ^(j), ξ₁ ^(j), ξ₂ ^(j), . . . , ξ_(n) _(j) ^(j) are (n_(j)+1)reference points along the j^(th) dimension. These are the primaryreference points as they lie along the j^(th) dimensional axis.

[0046] i: Ranges from one to total number of reference points ‘q’.

[0047] As k_(j) independently varies from 0 to n_(j) for each LagrangeInterpolation Polynomial,

q=(n ₁+1)*(n ₂+1)*(n ₃+1)* . . . *(n _(j)+1)* . . . *(n _(p)+1)  (6)

[0048] Thus, the co-ordinates for a reference point ‘i’, correspondingto each dimension, are given as (k₁, k₂, . . . , k_(j), . . . , k_(p)).

[0049] The reference points, which lie along the dimensional axes, arespecial points and are referred to as “primary reference points”. Interms of co-ordinates, a reference point is along the dimensional axisif one and only one of its co-ordinates (k_(j)) has a non-zero value andall other co-ordinates is zero.

[0050] A weight variable with values constrained between zero and unityis associated with each reference point. Therefore, the total number ofweights is the same as the total number of reference points ‘q’. Aweight value at a reference point, in the context of this invention, isconsidered to be representative of the belief value in the cause.Weights corresponding to primary reference points are chosen as primaryor independent weights and the remaining weights—referred to assecondary weights—are dependent on, and expressed as a linearcombination of primary weights. This dramatically reduces the number ofunknown variables within the network, thereby reducing the amount oftraining data required to a practical level as compared withconventional techniques. For example, for a ‘p’ dimensional problem, thenumber of primary weights is $\begin{matrix}{\left\lbrack {\left( {{\sum\limits_{j = 1}^{p}n_{j}} + 1} \right) - \left( {p - 1} \right)} \right\rbrack.} & (7)\end{matrix}$

[0051] (7).

[0052] During the learning or training process, the optimal values forprimary weights (constrained between zero and one) and coefficients usedin the linear combination expression are determined based on any userdefined method including generally available optimisation principles.The secondary weights are also constrained between zero and one.

[0053] The belief value in the occurrence of a cause, based on the knownbelief values (ξ^(j), j=1 to p) quantifying the strength of ‘p’associated effects, is given by the following equation: $\begin{matrix}{\text{The~~belief~~value~~in~~the~~cause} = {\sum\limits_{i = 1}^{q}{w_{i}{l_{i}\left( {\xi^{1},\xi^{2},\ldots \quad,\xi^{p}} \right)}}}} & (8)\end{matrix}$

[0054] where,

[0055] q: Total number of reference points.

[0056] l_(i)(ξ¹, ξ², . . . ξ^(p)) is given by equation (4).

[0057] w_(i): Weight variable associated with the i^(th) referencepoint.

[0058] In an embodiment of the present invention, a secondary weight isdefined as a linear combination of primary weights. However, thefollowing particular cases of linear combination may also be used inother preferred embodiments of the present invention.

[0059] 1. A secondary weight value associated with a reference point isa linear combination of those primary weight values, which areassociated with primary reference points corresponding to theco-ordinates of the said reference point.

[0060] 2. A secondary weight value associated with a reference point isa constant multiplied by the average of primary weight values, which areassociated with primary reference points corresponding to theco-ordinates of the said reference point.

[0061] A Numerical Example of an Exemplary Embodiment of the PresentInvention

[0062] Two effects, ξ¹ and ξ², are associated with a cause c. Therefore,two-dimensional Lagrange Interpolation Polynomials l_(i)(ξ¹, ξ²) will beused for defining the hypersurface. Quadratic relationship is assumedbetween belief values for the effect ξ¹ and the cause c, and also forthe effect ξ² and the cause c.

[0063] For this example, the belief value in the occurrence of cause cis calculated for a belief value in the first effect ξ¹ equal to 0.5 anda belief value in the second effect ξ² equal to −0.5.

[0064] Numbers 1 to 9 in FIG. 4 denote equidistant reference points. Asa result of quadratic relationship, (ξ¹ and ξ² equal to 2) threeequidistant reference points are used along each dimension. Usingequation (6), it can be seen that the total number of reference pointsis 9.

[0065] Using equation (7), it can be seen that total number of primaryreference points is 5. These points are also indicated in the followingtable (Table 1), which shows the co-ordinates of all nine referencepoints in various forms. TABLE 1 Reference Coordinates in terms PointsCoordinates in Coordinates in of actual numerical (i = 1 to q) terms ofk_(j) terms of ξ^(j) values for ξ^(j)'s 1. (0,0) Primary (ξ₀ ¹,ξ₀ ²)(−1,−1) 2. (1,0) Primary (ξ₁ ¹,ξ₀ ²) (0,−1) 3. (2,0) Primary (ξ₂ ¹,ξ₀ ²)(+1,−1) 4. (0,1) Primary (ξ₀ ¹,ξ₁ ²) (−1,0) 5. (1,1) Secondary (ξ₁ ¹,ξ₁²) (0,0) 6. (2,1) Secondary (ξ₂ ¹,ξ₁ ²) (1,0) 7. (0,2) Primary (ξ₀ ¹,ξ₂²) (−1,+1) 8. (1,2) Secondary (ξ₁ ¹,ξ₂ ²) (0,+1) 9. (2,2) Secondary (ξ₂¹,ξ₂ ²) (+1,+1)

[0066] Weights associated with primary reference points 1, 2, 3, 4 and 7are primary weights and are also independent parameters. The secondaryweight values at locations 5, 6, 8 and 9 are expressed as linearcombination of primary weights and in particular $\begin{matrix}{w_{5} = \frac{C\left( {w_{2} + w_{4}} \right)}{2}} & (9) \\{w_{6} = \frac{C\left( {w_{3} + w_{4}} \right)}{2}} & (10) \\{w_{8} = {\frac{C\left( {w_{2} + w_{7}} \right)}{2}\quad \text{and}}} & (11) \\{w_{9} = {\frac{C\left( {w_{3} + w_{7}} \right)}{2}.}} & (12)\end{matrix}$

[0067] Thus, the independent parameters which are passed on to theoptimisation algorithm during a learning process are w₁, w₂, w₃, w₄, w₇and constant C.

[0068] The belief value, in the extent of the occurrence of cause, for agiven belief value ξ¹ equal to 0.5 (representing the strength of effectξ¹) and a given belief value ξ² equal to −0.5 (representing the strengthof effect ξ²) is calculated as:$\text{The~~belief~~value~~in~~the~~occurence~~of~~cause} = {\sum\limits_{i = 1}^{9}{w_{i}{l_{i}\left( {0.5,{- 0.5}} \right)}}}$

[0069] Using equations (4) and (5), Lagrange Interpolation Polynomialsare constructed and then evaluated at (0.5, −0.5) at all referencepoints.

[0070] Lagrange Interpolation Polynomial for reference point 1:$\begin{matrix}{{l_{1}\left( {\xi^{1},\xi^{2}} \right)} = {{l_{0}^{2}\left( \xi^{1} \right)}*{l_{0}^{2}\left( \xi^{2} \right)}}} \\{= {\left( {\frac{\xi^{1} - \xi_{1}^{1}}{\xi_{0}^{1} - \xi_{1}^{1}}*\frac{\xi^{1} - \xi_{2}^{1}}{\xi_{0}^{1} - \xi_{2}^{1}}} \right)*\left( {\frac{\xi^{2} - \xi_{1}^{2}}{\xi_{0}^{2} - \xi_{1}^{2}}*\frac{\xi^{2} - \xi_{2}^{2}}{\xi_{0}^{2} - \xi_{2}^{2}}} \right)}} \\{= {\frac{1}{4}\left( \xi^{1} \right)\left( {\xi^{1} - 1} \right)\left( \xi^{2} \right)\left( {\xi^{2} - 1} \right)}} \\{{l_{1}\left( {0.5,{- 0.5}} \right)} = {- 0.0469}}\end{matrix}$

[0071] Lagrange Interpolation Polynomial for reference point 2:$\begin{matrix}{{l_{2}\left( {\xi^{1},\xi^{2}} \right)} = {{l_{1}^{2}\left( \xi^{1} \right)}*{l_{0}^{2}\left( \xi^{2} \right)}}} \\{= {\left( {\frac{\xi^{1} - \xi_{0}^{1}}{\xi_{1}^{1} - \xi_{1}^{1}}*\frac{\xi^{1} - \xi_{2}^{1}}{\xi_{1}^{1} - \xi_{2}^{1}}} \right)*\left( {\frac{\xi^{2} - \xi_{1}^{2}}{\xi_{0}^{2} - \xi_{1}^{2}}*\frac{\xi^{2} - \xi_{2}^{2}}{\xi_{0}^{2} - \xi_{2}^{2}}} \right)}} \\{= {{- \frac{1}{2}}\left( {\xi^{1} + 1} \right)\left( {\xi^{1} - 1} \right)\left( \xi^{2} \right)\left( {\xi^{2} - 1} \right)}} \\{{l_{2}\left( {0.5,{- 0.5}} \right)} = 0.2813}\end{matrix}$

[0072] Lagrange Interpolation Polynomial for reference point 3:$\begin{matrix}{{l_{3}\left( {\xi^{1},\xi^{2}} \right)} - {{l_{2}^{2}\left( \xi^{1} \right)}*{l_{0}^{2}\left( \xi^{2} \right)}}} \\{= {\left( {\frac{\xi^{1} - \xi_{0}^{1}}{\xi_{2}^{1} - \xi_{0}^{1}}*\frac{\xi^{1} - \xi_{1}^{1}}{\xi_{2}^{1} - \xi_{1}^{1}}} \right)*\left( {\frac{\xi^{2} - \xi_{1}^{2}}{\xi_{0}^{1} - \xi_{1}^{2}}*\frac{\xi^{2} - \xi_{2}^{2}}{\xi_{0}^{2} - \xi_{2}^{1}}} \right)}} \\{= {\frac{1}{4}\left( {\xi^{1} + 1} \right)\left( \xi^{1} \right)\left( \xi^{2} \right)\left( {\xi^{2} - 1} \right)}} \\{{l_{3}\left( {0.5,{- 0.5}} \right)} = 0.1406}\end{matrix}$

[0073] Lagrange Interpolation Polynomial for reference point 4:$\begin{matrix}{{l_{4}\left( {\xi^{1},\xi^{2}} \right)} = {{l_{0}^{2}\left( \xi^{1} \right)}*{l_{1}^{2}\left( \xi^{2} \right)}}} \\{= {\left( {\frac{\xi^{1} - \xi_{1}^{1}}{\xi_{0}^{1} - \xi_{1}^{1}}*\frac{\xi^{1} - \xi_{2}^{1}}{\xi_{0}^{1} - \xi_{2}^{1}}} \right)*\left( {\frac{\xi^{2} - \xi_{0}^{2}}{\xi_{1}^{2} - \xi_{0}^{2}}*\frac{\xi^{2} - \xi_{2}^{2}}{\xi_{1}^{2} - \xi_{2}^{2}}} \right)}} \\{= {{- \frac{1}{2}}\left( \xi^{1} \right)\left( {\xi^{1} - 1} \right)\left( {\xi^{2} + 1} \right)\left( {\xi^{2} - 1} \right)}} \\{{l_{4}\left( {0.5,{- 0.5}} \right)} = {- 0.0938}}\end{matrix}$

[0074] Lagrange Interpolation Polynomial for reference point 5:$\begin{matrix}{{l_{5}\left( {\xi^{1},\xi^{2}} \right)} = {{l_{1}^{2}\left( \xi^{1} \right)}*{l_{1}^{2}\left( \xi^{2} \right)}}} \\{= {\left( {\frac{\xi^{1} - \xi_{0}^{1}}{\xi_{1}^{1} - \xi_{0}^{1}}*\frac{\xi^{1} - \xi_{2}^{1}}{\xi_{1}^{1} - \xi_{2}^{1}}} \right)*\left( {\frac{\xi^{2} - \xi_{0}^{2}}{\xi_{1}^{2} - \xi_{0}^{2}}*\frac{\xi^{2} - \xi_{2}^{2}}{\xi_{1}^{2} - \xi_{2}^{2}}} \right)}} \\{= {\left( {\xi^{1} + 1} \right)\left( {\xi^{1} - 1} \right)\left( {\xi^{2} + 1} \right)\left( {\xi^{2} - 1} \right)}} \\{{l_{5}\left( {0.5,{- 0.5}} \right)} = 0.5625}\end{matrix}$

[0075] Lagrange Interpolation Polynomial for reference point 6:$\begin{matrix}{{l_{6}\left( {\xi^{1},\xi^{2}} \right)} = {{l_{2}^{2}\left( \xi^{1} \right)}*{l_{1}^{2}\left( \xi^{2} \right)}}} \\{= {\left( {\frac{\xi^{1} - \xi_{0}^{1}}{\xi_{2}^{1} - \xi_{0}^{1}}*\frac{\xi^{1} - \xi_{1}^{1}}{\xi_{2}^{1} - \xi_{1}^{1}}} \right)*\left( {\frac{\xi^{2} - \xi_{0}^{2}}{\xi_{1}^{2} - \xi_{0}^{2}}*\frac{\xi^{2} - \xi_{2}^{2}}{\xi_{1}^{2} - \xi_{2}^{2}}} \right)}} \\{= {{- \frac{1}{2}}\left( {\xi^{1} + 1} \right)\left( \xi^{1} \right)\left( {\xi^{2} + 1} \right)\left( {\xi^{2} - 1} \right)}} \\{{l_{6}\left( {0.5,{- 0.5}} \right)} = 0.2813}\end{matrix}$

[0076] Lagrange Interpolation Polynomial for reference point 7:$\begin{matrix}{{l_{7}\left( {\xi^{1},\xi^{2}} \right)} = {{l_{0}^{2}\left( \xi^{1} \right)}*{l_{2}^{2}\left( \xi^{2} \right)}}} \\{= {\left( {\frac{\xi^{1} - \xi_{1}^{1}}{\xi_{0}^{1} - \xi_{1}^{1}}*\frac{\xi^{1} - \xi_{2}^{1}}{\xi_{0}^{1} - \xi_{2}^{1}}} \right)*\left( {\frac{\xi^{2} - \xi_{0}^{2}}{\xi_{2}^{2} - \xi_{0}^{2}}*\frac{\xi^{2} - \xi_{1}^{2}}{\xi_{2}^{2} - \xi_{1}^{2}}} \right)}} \\{= {\frac{1}{4}\left( \xi^{1} \right)\left( {\xi^{1} - 1} \right)\left( {\xi^{2} + 1} \right)\left( \xi^{2} \right)}} \\{{l_{7}\left( {0.5,{- 0.5}} \right)} = 0.0156}\end{matrix}$

[0077] Lagrange Interpolation Polynomial for reference point 8:$\begin{matrix}{{l_{8}\left( {\xi^{1},\xi^{2}} \right)} = {{l_{1}^{2}\left( \xi^{1} \right)}*{l_{2}^{2}\left( \xi^{2} \right)}}} \\{= {\left( {\frac{\xi^{1} - \xi_{0}^{1}}{\xi_{1}^{1} - \xi_{0}^{1}}*\frac{\xi^{1} - \xi_{2}^{1}}{\xi_{1}^{1} - \xi_{2}^{1}}} \right)*\left( {\frac{\xi^{2} - \xi_{0}^{2}}{\xi_{2}^{2} - \xi_{0}^{2}}*\frac{\xi^{2} - \xi_{1}^{2}}{\xi_{2}^{2} - \xi_{1}^{2}}} \right)}} \\{= {{- \frac{1}{2}}\left( {\xi^{1} + 1} \right)\left( {\xi^{1} - 1} \right)\left( {\xi^{2} + 1} \right)\left( \xi^{2} \right)}} \\{{l_{8}\left( {0.5,{- 0.5}} \right)} = {- 0.0938}}\end{matrix}$

[0078] Lagrange Interpolation Polynomial for reference point 9:$\begin{matrix}{{l_{9}\left( {\xi^{1},\xi^{2}} \right)} = {{l_{2}^{2}\left( \xi^{1} \right)}*{l_{1}^{2}\left( \xi^{2} \right)}}} \\{= {\left( {\frac{\xi^{1} - \xi_{0}^{1}}{\xi_{2}^{1} - \xi_{0}^{1}}*\frac{\xi^{1} - \xi_{1}^{1}}{\xi_{2}^{1} - \xi_{1}^{1}}} \right)*\left( {\frac{\xi^{2} - \xi_{0}^{2}}{\xi_{2}^{2} - \xi_{0}^{2}}*\frac{\xi^{2} - \xi_{1}^{2}}{\xi_{2}^{2} - \xi_{1}^{2}}} \right)}} \\{= {\frac{1}{4}\left( {\xi^{1} + 1} \right)\left( \xi^{1} \right)\left( {\xi^{2} + 1} \right)\left( \xi^{2} \right)}} \\{{l_{9}\left( {0.5,{- 0.5}} \right)} = {- 0.0469}}\end{matrix}$

[0079] If w₁=0.0084, w₂=0.1972, w₃=0.4179, w₄=0.1924, w₇=0.7359 andC=1.5656, then using equations (9), (10), (11) and (12) w₅=0.3050,w₆=0.4778, w₈=0.7304 and w₉=0.9032. $\begin{matrix}{{{Belief}\quad {value}\quad {in}\quad {the}\quad {cause}} = {\sum\limits_{i = 1}^{9}\quad {w_{i}{l_{i}\left( {0.5,{- 0.5}} \right)}}}} \\{= 0.3024}\end{matrix}$

[0080] During a learning or training phase, the predicted belief value(0.3024) is compared with a known belief value in the training data fileto calculate the error. Any of the known optimisation method is used tocalculate new values of primary weights and coefficients used in thelinear combination equations. The new belief value in the cause is againcalculated and the process is repeated until a user defined criterion oferror minimisation is achieved.

[0081] The optimal values of primary weights and coefficients in thelinear combination equations are stored and used for futureapplications.

[0082] Embodiments of the present invention has been described above byway of example only and it will be appreciated by a person skilled inthe art that modifications and variations can be made to the describedembodiments without departing from the scope of the invention.

1. Apparatus for determining a likely cause of one or more effects, theapparatus comprising means for receiving and/or accessing training datarelating to previously-identified relationships between one or morecauses and one or more effects, means for identifying a trend in saidrelationships and in the form of shape functions or polynomials whichdefine quantified mappings between said one or more causes and one ormore effects, each of said shape functions or polynomials beingassociated with each of a plurality of respective reference points, atleast some of which have a weight value assigned thereto at least one ofsaid weight values being an independent variable (“primary weight”) andat least one of said weight values being dependent on one or more ofsaid primary weights (“secondary weight”), and means for determining oneor more likely causes of one or more given effects using said mappings.2. Apparatus according to claim 1, comprising a neural networkarrangement being capable of providing outputs in the form ofquantitative cause and effect relationship values.
 3. A method ofdetermining a likely cause of one or more effects, the method comprisingthe steps of receiving and/or accessing quantifying data relating topreviously-identified relationships between one or more causes and oneor more effects, identifying a trend in said relationships in the formof shape functions or polynomials which define quantified mappingsbetween said one or more causes and one or more effects, each said shapefunctions or polynomials being associated with each of a plurality ofrespective reference points, at least some of which have a weight valueassigned thereto, at least one of said weight values being anindependent variable (“primary weight”) and at least one of said weightvalues being dependent on at least one of said primary weights(“secondary weight”), and determining one or more likely causes of oneor more given effects using said mappings.
 4. Apparatus according toclaim 1, wherein the trend is defined in terms of respective shapefunctions which produce a hyper-surface representative of the trainingdata in terms of quantitative cause and effect relationship values. 5.Apparatus according to claim 1, including means for producing amulti-dimensional hyper-surface representing a trend in the trainingdata, the number of dimensions of the hyper-surface being equal to thenumber of input nodes of a neural network.
 6. Apparatus according toclaim 5, wherein the neural network includes one or more layers eachcomprising one or more hidden nodes between said input and output nodes.7. Apparatus according to claim 5, wherein shape functions are definedat each input and output node of the neural network, each shape functionhaving output values between 0 and
 1. 8. Apparatus according to claim 7,wherein the value of the shape function at its respective node is 1, and0 elsewhere.
 9. Apparatus according to claim 5, wherein at least eachinput and output node in the neural network has a weight value assignedto it.
 10. Apparatus according to claim 9, wherein the value of a weightis between −1 and +1.
 11. Apparatus according to claim 9, herein theoutput of the network is preferably also between −1 and +1. 12.Apparatus according to claim 1, in which each secondary weight isexpressed as a linear combination of two or more primary weights. 13.Apparatus according to claim 4, wherein the hyper-surface is createdusing one or more multi-dimensional (linear, quadratic, cubic or higherorder) polynomial equations.
 14. Apparatus according to claim 1, whereinthe training data is made up of one or more training files, the or eachtraining file comprising an input vector and its corresponding desiredvector.
 15. (canceled)
 16. (canceled)
 17. Method according to claim 3,wherein the trend is defined in terms of respective shape functionswhich produce a hyper-surface representative of the training data interms of quantitative cause and effect relationship values.
 18. Methodaccording to claim 17, wherein the hyper-surface is created using one ormore multi-dimensional (linear, quadratic, cubic or higher order)polynomial equations.
 19. Method according to claim 3, wherein thetraining data is made up of one or more training files, the or eachtraining file comprising an input vector and its corresponding desiredvector.